High-speed large-capacity optical fiber transmission systems are directions of development for future optical communications, in which high-order quadrature amplitude modulation (QAM) formats combined with coherent receiving technique provide a promising solution. On the one hand, the use of QAM formats may improve the spectral efficiency and lower the demand for the electro-optic bandwidth; and on the other hand, the coherent receiving technique may allow for overcoming the signal transmission impairments with the powerful digital signal processing (DSP) techniques while improving the receiving sensitivity.
Since the existence of the laser phase noise, a carrier phase estimator is an indispensable device in a typical digital coherent receiver. In addition, in a practical application, it is often expected to use a feed-forward blind phase estimation method to avoid the use of a training sequence, thereby improving the transmission efficiency of information. However, the tolerance of a QAM signal to a laser phase noise significantly decreases as the modulation order increases, and since the constellation points of a high-order QAM signal are more dense, the conventional feed-forward blind carrier phase estimation algorithms are hard to be expended to a high-order QAM signal (documentations[1-4]), this brings forward a high requirement on the feed-forward blind carrier phase estimation modules.
Among various proposed feed-forward blind carrier phase estimation algorithms, a feed-forward algorithm (documentations or references[5-7]) based on blind phase search (BPS) (note: in different references, the English names of such an algorithm are not completely the same and have no definite Chinese names, which are collectively referred to as an algorithm based on blind phase search in this application) has many advantages, such as better phase noise tolerance, parallel processing feasibility, and universality to all-order QAM formats, etc. The principle of such an algorithm is relatively simple, but its implementation complexity is very high. An effective solution is to expand a single-stage phase estimation module to a multi-stage module to reduce the number of phase angles needed in the phase search, thereby reduce the complexity.
Several multi-stage phase estimation algorithms were proposed in documentations or references[8-11], in which the complexity was reduced by a factor of 1.5 to 3. Where the algorithm proposed in documentations[10] may be used for reference. In the two-stage phase estimation configuration proposed in documentations[10], each stage is based on the phase search algorithm, where the former stage may be regarded as a coarse search of the latter, and the latter stage is a fine search conducted on the basis of the former stage estimated phase, thereby reducing the number of the phase angles for the phase search while ensuring the precision of the phase estimation. However, since average time window with identical lengths are used in its two-stage configuration, the number of the phase angles needed in the first stage is still very large due to the influence of the pattern effect, and thus, the reduction of its complexity is still limited.
Following documentations or references may be helpful to the understanding of the embodiments and the conventional technologies listed below, which are incorporated herein by reference as they are completely set forth in this text.    [1] R. Noé, “Phase noise tolerant synchronous QPSK/BPSK baseband-type intradyne receiver concept with feed-forward carrier recovery,” J. Lightw. Technol., vol. 23, no. 2, pp. 802-808, February 2005.    [2] H. Louchet, K. Kuzmin, and A. Richter, “Improved DSP algorithms for coherent 16-QAM transmission,” PaperTu.1.E.6, in Proc. ECOC2008, Brussels, Belgium, Sep. 21-25, 2008.    [3] M. Seimetz, “Laser linewidth limitations for optical systems with high-order modulation employing feedforward digital carrier phase estimation,” PaperOTuM2, in Proc. OFC2008, San Diego, Calif., Feb. 24-28, 2008.    [4] I. Fatadin, D. Ives, and S. J. Savory, “Laser linewidth tolerance for 16QAM coherent optical systems using QPSK partitioning,” IEEE Photon. Technol. Lett., vol. 22, no. 9, pp. 631-633, May 2010.    [5] S. K. Oh and S. P. Stapleion, “Blind phase recovery using finite alphabet properties in digital communications,” Electronics Letters, vol. 33, no. 3, pp. 175-176, January 1997.    [6] F. Rice, B. Cowley, B. Moran, and M. Rice, “Cramér-Rao lower bounds for QAM phase and frequency estimation,” IEEE Transactions on Communications, vol. 49, no. 9, pp. 1582-1591, September 2001.    [7] T. Pfau, S. Hoffmann, and R. Noe, “Hardware-efficient coherent digital receiver concept with feed-forward carrier recovery for M-QAM constellations,” Journal of Lightwave Technology, vol. 27, no. 8, pp. 989-999, Apr. 15, 2009.    [8] T. Pfau, and R. Noe, “Phase-noise-tolerant two-stage carrier recovery concept for higher order QAM formats,” IEEE Journal of Selected Topics on Quantum Electronics, vol. 16, no. 5, pp. 1210-1216, 2010.    [9] X. Zhou, “An improved feed-forward carrier recovery algorithm for coherent receivers with M-QAM modulation format,” IEEE Photonics Technology Letters, vol. 22, no. 14, pp. 1051-1053, July 2010.    [10] X. Li, Y. Cao, S. Yu, W. Gu, and Y. Ji, “A Simplified Feed-Forward Carrier Recovery Algorithm for Coherent Optical QAM System,” Journal of Lightwave Technology, vol. 29, no. 5, pp. 801-807, March 2011.    [11] Q. Zhuge, C. Chen, and D. V. Plant, “Low computation complexity two-stage feedforward carrier recovery algorithm for M-QAM,” Paper OMJ5, presented in OFC2011, Los Angeles, Calif., March 2011.